It seems there are multiple causes for this error, both hardware and software related. Is it only Wii party? or other discs too? If it is only Wii party then it is probably the disc. It may be badly scratched.
First try cleaning the disc. There are two ways to go about this. Either go and buy a dvd cleaning kit and follow the instructions, or use my toothpaste method.
Toothpaste is lightly abrasive, and can polish scratches out of discs if not too badly damaged. Wet the disc first, then start to gently rub toothpaste around it in little swirls until the whole disc is covered. Rinse the disc off with water and a cotton cloth, then dry with cotton or glass cleaning material.
Try the disc again.
If the problem is occurring with other discs as well; then the problem is probably hardware related. The safest course of action in this case is to contact Nintendo support.